Abstract
Let \(K_{n}\) denote a complete graph on n vertices, where n is even. In recent works, geometric methods (finite projective planes, regular gons in the Euclidean plane) have been developed in the study of one-factorizations of \(K_{n}\), but a geometric viewpoint for perfect one-factorizations still remains unknown. In this note we apply the Lee metric to construct a one-factorization of \(K_{2p}\) for an odd number p, which is isomorphic to those factorizations by Anderson and by Nakamura. Moreover, this construction is perfect for an odd prime p. By combining our approach with previous results, some known maximum distance separable codes in the Hamming space can be derived from the Lee metric.
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The authors are grateful to the anonymous referee for pointing out updated references and several suggestions.
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The authors have no relevant financial or non-financial interests to disclose. All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by the both authors. All authors read and approved the final manuscript.
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P.H Perondi was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
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Perondi, P.H., Monte Carmelo, E.L. Perfect One-Factorizations Arising from the Lee Metric. Graphs and Combinatorics 39, 8 (2023). https://doi.org/10.1007/s00373-022-02603-x
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DOI: https://doi.org/10.1007/s00373-022-02603-x